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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{exotic smooth structure} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry} [[!include synthetic differential geometry - contents]] \hypertarget{topology}{}\paragraph*{{Topology}}\label{topology} [[!include topology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{existence_and_examples}{Existence and Examples}\dotfill \pageref*{existence_and_examples} \linebreak \noindent\hyperlink{no_exotic_smooth_structure_in_dimensions_}{No exotic smooth structure in dimensions $\leq 3$}\dotfill \pageref*{no_exotic_smooth_structure_in_dimensions_} \linebreak \noindent\hyperlink{no_exotic_euclidean_space_away_from_dimension_4}{No exotic Euclidean space away from dimension 4}\dotfill \pageref*{no_exotic_euclidean_space_away_from_dimension_4} \linebreak \noindent\hyperlink{ExoticEuclideal4Space}{Exotic Euclidean 4-space}\dotfill \pageref*{ExoticEuclideal4Space} \linebreak \noindent\hyperlink{Exotic4Spheres}{Exotic 4-spheres?}\dotfill \pageref*{Exotic4Spheres} \linebreak \noindent\hyperlink{Exotic7Sphere}{Exotic 7-sphere}\dotfill \pageref*{Exotic7Sphere} \linebreak \noindent\hyperlink{ExoticNSpheresForNGreaterThanFour}{Exotic $n$-spheres for $n \geq 5$}\dotfill \pageref*{ExoticNSpheresForNGreaterThanFour} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{for_the_mathematical_theory}{For the mathematical theory}\dotfill \pageref*{for_the_mathematical_theory} \linebreak \noindent\hyperlink{ReferencesForApplicationsToPhysics}{For applications to physics}\dotfill \pageref*{ReferencesForApplicationsToPhysics} \linebreak \noindent\hyperlink{general_relativity}{General relativity}\dotfill \pageref*{general_relativity} \linebreak \noindent\hyperlink{generation_of_source_terms_fields}{Generation of source terms (fields)}\dotfill \pageref*{generation_of_source_terms_fields} \linebreak \noindent\hyperlink{quantum_field_theory}{Quantum (field) theory}\dotfill \pageref*{quantum_field_theory} \linebreak \noindent\hyperlink{ReferencesInStringTheory}{String theory}\dotfill \pageref*{ReferencesInStringTheory} \linebreak \noindent\hyperlink{cosmology}{Cosmology}\dotfill \pageref*{cosmology} \linebreak \noindent\hyperlink{quantum_gravity}{Quantum gravity}\dotfill \pageref*{quantum_gravity} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} An \emph{exotic smooth structure} is, roughly speaking, a [[smooth structure]] on a [[topological manifold]] $X$ which makes the resulting [[smooth manifold]] be [[diffeomorphism|non-diffeomorphic]] to the smooth manifold given by some evident `standard' smooth structure on $X$. Mostly the term is used for smooth structures on [[Euclidean space]] $\mathbb{R}^n$ and on the [[n-spheres]] $S^n$, for $n \in \mathbb{N}$. The standard smooth structure on $\mathbb{R}^n$ is exhibited by the identity [[atlas]], and the standard smooth structure on $S^n$ is that given by the atlas by the two [[hemisphere]] as given by [[stereographic projection]]. For special values of $n$ there may exist smooth structure not equivalent to these. They are the \emph{exotic smooth structures}. A classification of [[smooth manifold|smooth]], [[piecewise-linear manifold|PL]] and [[topological manifold|topological]] structures on manifolds in dimension 5 and higher, in terms of various groups from [[algebraic topology]] (many not known) was established by \hyperlink{KirbSieb}{Kirby and Siebenmann (1977)} using [[obstruction theory]]. \hypertarget{existence_and_examples}{}\subsection*{{Existence and Examples}}\label{existence_and_examples} \hypertarget{no_exotic_smooth_structure_in_dimensions_}{}\subsubsection*{{No exotic smooth structure in dimensions $\leq 3$}}\label{no_exotic_smooth_structure_in_dimensions_} \hyperlink{Rad}{Rado (1925)} proved that in dimension 2 there are no exotic differentiable structures (or the uniqueness of the standard structure). The classification of 1-dimensional manifolds and the uniqueness of the smooth structure can be found in the Appendix of \hyperlink{Milnor1965b}{Milnor (1965b)}. \hyperlink{Moise}{Moise (1952)} proved that in dimension 3 there are no exotic differentiable structures, or to put in another way, 3-dimensional differentiable manifolds which are [[homeomorphism|homeomorphic]] are [[diffeomorphism|diffeomorphic]]. In this way the 3-sphere $S^3$ inherits a unique differentiable structure, no matter which $\mathbb{R}^4$ it is considered to be embedded in. \hypertarget{no_exotic_euclidean_space_away_from_dimension_4}{}\subsubsection*{{No exotic Euclidean space away from dimension 4}}\label{no_exotic_euclidean_space_away_from_dimension_4} There exists a unique smooth structure on the [[Euclidean space]] $\mathbb{R}^n$ for $n\neq 4$ (\hyperlink{Stallings62}{Stallings 1962}). \hypertarget{ExoticEuclideal4Space}{}\subsubsection*{{Exotic Euclidean 4-space}}\label{ExoticEuclideal4Space} There exists uncountably many exotic smooth structures on the [[Euclidean space]] $\mathbb{R}^4$ of dimension 4 (\hyperlink{Gompf}{Gompf 1985}, \hyperlink{FreedTay}{Freedman/Taylor 1986}, \hyperlink{Taubes}{Taubes 1987}). See also at \emph{[[exotic R{\tt \symbol{94}}4]]}. There is a unique maximal exotic $\mathbb{R}^4$ into which all other `versions' of $\mathbb{R}^4$ smoothly embed as open subsets (Freedman/Taylor 1986, \hyperlink{DeMFreed}{DeMichelis/Freedman 1992}). There are two classes of exotic $\mathbb{R}^4$`s: large and small. A large exotic $\mathbb{R}^4$ cannot be embedded in the 4-sphere $S^4$ (Gompf 1985, Taubes 1987) whereas a small exotic $\mathbb{R}^4$ admits such an embedding (DeMichelis/Freedman 1992): \begin{itemize}% \item A \emph{large exotic $\mathbb{R}^4$} is constructed by using the failure to smoothly split a smooth 4-manifold (the [[K3 surface]] for instance) as a [[connected sum]] of some factors (where a topological splitting exits). \item The \emph{small exotic $\mathbb{R}^4$} (or \emph{ribbon $\mathbb{R}^4$}) is constructed by using the failure of the smooth [[h-cobordism theorem]] in dimension 4 (\hyperlink{Donaldson1}{Donaldson 1987}, \hyperlink{Donaldson2}{1990}). \hyperlink{BizGompf}{Bizaca and Gompf (1996)} are able to present an infinite handle body of a small exotic $\mathbb{R}^4$ which serve as a coordinate representation. \end{itemize} \hypertarget{Exotic4Spheres}{}\subsubsection*{{Exotic 4-spheres?}}\label{Exotic4Spheres} It is open whether the [[4-sphere]] admits an exotic smooth structure. See (\hyperlink{FreedmanGompfMorrisonWalker09}{Freedman-Gompf-Morrison-Walker 09} for review). \hypertarget{Exotic7Sphere}{}\subsubsection*{{Exotic 7-sphere}}\label{Exotic7Sphere} \hyperlink{Milnor1956}{Milnor (1956)} gave the first examples of exotic smooth structures on the [[7-sphere]], finding at least seven. The [[exotic 7-spheres]] constructed in \hyperlink{Milnor1956}{Milnor 1956} are all examples of [[fibre bundles]] over the [[4-sphere]] $S^4$ with [[fibre]] the [[3-sphere]] $S^3$, with [[structure group]] the [[special orthogonal group]] [[SO(4)]] (see also at \emph{[[8-manifold]]} the section \emph{\href{8-manifold#ExoticBoundary7Spheres}{With exotic boundary 7-spheres}}): By the classification of bundles on spheres via the [[clutching construction]], these correspond to [[homotopy classes]] of maps $S^3 \to SO(4)$, i.e. elements of $\pi_3(SO(4))$. From the table at \href{orthogonal+group#HomotopyGroups}{orthogonal group -- Homotopy groups}, this latter group is $\mathbb{Z}\oplus\mathbb{Z}$. Thus any such bundle can be described, up to [[isomorphism]], by a [[pair]] of [[integers]] $(n,m)$. When $n+m=1$, then one can show there is a [[Morse function]] with exactly two [[critical points]] on the total space of the bundle, and hence this 7-manifold is [[homeomorphic]] to a sphere. The [[fractional first Pontryagin class]] $\frac{p_1}{2} \in H^4(S^4) \simeq \mathbb{Z}$ of the bundle is given by $n-m$. Milnor constructs, using [[cobordism]] theory and [[Hirzebruch's signature theorem]] for 8-manifolds, a mod-7 diffeomorphism invariant of the manifold, so that it is standard 7-sphere precisely when $\frac{p_1}{2}^2 -1 = 0 (mod 7)$. By using the [[connected sum]] operation, the set of smooth, non-diffeomorphic structures on the $n$-sphere has the structure of an [[abelian group]]. For the [[7-sphere]], it is the [[cyclic group]] $Z/{28}$ and Brieskorn (1966) found the generator $\Sigma$ so that $\underbrace{\Sigma\#\cdots\#\Sigma}_28$ is the standard sphere. For review see for instance (\hyperlink{Kreck10}{Kreck 10, chapter 19}, \hyperlink{McEnroe15}{McEnroe 15}). For more see at \emph{[[exotic 7-sphere]]}. From the point of view of [[M-theory on 8-manifolds]], these [[8-manifolds]] $X$ with (exotic) [[7-sphere]] [[boundaries]] correspond to [[near horizon limits]] of [[black brane|black]] [[M2 brane]] spacetimes $\mathbb{R}^{2,1} \times X$, where the [[M2-branes]] themselves would be sitting at the center of the [[7-spheres]] (if that were included in the spacetime, see also [[Dirac charge quantization]]). (\href{M-theory+on+8-manifolds#MorrisonPlesser99}{Morrison-Plesser 99, section 3.2}) $\backslash$linebreak \hypertarget{ExoticNSpheresForNGreaterThanFour}{}\subsubsection*{{Exotic $n$-spheres for $n \geq 5$}}\label{ExoticNSpheresForNGreaterThanFour} Via the celebrated [[h cobordism theorem]] of Smale (\hyperlink{Smale}{Smale 1962}, \hyperlink{Milnor1965a}{Milnor 1965}) one gets a relation between the number of smooth structures on the $n$-sphere $S^n$ (for $n \geq 5$) and the number of [[isotopy]] classes $\pi_0 (Diff(S^{n-1}))$ of the [[equator]] $S^{n-1}$. Then \hyperlink{KervMil}{Kervaire and Milnor (1963)} proved that for each $n \geq 5$ there are only finitely many exotic smooth structures on the [[n-sphere]] $S^n$ (possibly none). By using the [[connected sum]] operation, the set of smooth, non-diffeomorphic structures on the $n$-sphere has the structure of an [[abelian group]]. The only odd-dimensional spheres with \emph{no} exotic smooth structure are the [[circle]] $S^1$, the [[3-sphere]] $S^3$, as well as $S^5$ and $S^{61}$ (\hyperlink{WangXu16}{Wang-Xu 16, corollary 1.13}) In the range $5 \leq n \leq 61$ the only $n$-spheres with \emph{no} exotic smooth structures are $S^5$, $S^6$, $S^{12}$, $S^{56}$ and $S^{61}$ (\hyperlink{WangXu16}{Wang-Xu 16, corollary 1.15}). It is conjectured that this exhausts in fact all examples of $n$-spheres without exotic smooth structure for $n \geq 5$ (\hyperlink{WangXu16}{Wang-Xu 16, conjecture 1.17}). \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[differential topology]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} See also \begin{itemize}% \item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Exotic_sphere}{Exotic sphere}} \end{itemize} \hypertarget{for_the_mathematical_theory}{}\subsubsection*{{For the mathematical theory}}\label{for_the_mathematical_theory} The first construction of exotic smooth structures was on the 7-[[sphere]] in \begin{itemize}% \item [[John Milnor]] (1956), ``On manifolds homeomorphic to the 7-sphere'', Annals of Mathematics (Annals of Mathematics) 64 (2): 399--405 \end{itemize} (\ldots{}) \begin{itemize}% \item [[Stephen Smale]], \emph{On the structure of manifolds}, Amer. J. of Math. 84 : 387-399 (1962) \item [[John Milnor]] (1965), \emph{Lectures on the h-cobordism theorem} (Princeton Univ. Press, Princeton) \item [[Michel Kervaire]], ; [[John Milnor]], (1963) ``Groups of homotopy spheres: I'', Ann. Math. 77, pp. 504 - 537. \item Kirby, R.; Siebenmann, L. (1977) \emph{Foundational essays on topological manifolds, smoothings, and triangulations}, Ann. Math. Studies (Princeton University Press, Princeton). \item John R. Stallings, \emph{The piecewise-linear structure of Euclidean space}, Proceedings of the Cambridge Philosophical Society 58: 481--488 (1962) (\href{http://www.maths.ed.ac.uk/~aar/papers/stallings2.pdf}{pdf}) \item Moise, Edwin E. (1952) ``Affine structures on 3-manifolds'', Ann. Math. 56, pp. 96-114 \item Freedman, Michael H.; Taylor, Laurence (1986) ``A universal smoothing of four-space'', J. Diff. Geom. 24, pp. 69-78 \item De Michelis, Stefano; Freedman, Michael H. (1992) ``Uncountably many exotic $\mathbb{R}^4$`s in standard 4-space'', J. Diff. Geom. 35, pp. 219-254. \item [[Simon Donaldson]] (1987) ``Irrationality and the h-cobordism conjecture'', J. Diff. Geom. 26, pp. 141-168. \item [[Simon Donaldson]], (1990) ``Polynomial invariants for smooth four manifolds'', Topology 29, pp. 257-315. \item Gompf, Robert (1985) ``An infinite set of exotic $\mathbb{R}^4$`s'', J. Diff. Geom. 21, pp. 283-300. \item Taubes, Clifford H. (1987) ``Gauge theory on asymptotically periodic 4-manifolds'', J. Diff. Geom. 25, pp. 363-430 \item Bizaca, Z.; Gompf, Robert (1996) ``Elliptic surfaces and some simple exotic $\mathbb{R}^4$`s'', J. Diff. Geom. 43, pp. 458-504. \item Rado, T. (1925) ``\"U{}ber den Begriff der Riemannschen Fl\"a{}che'' , Acta Litt. Scient. Univ. Szegd 2, pp. 101-121 \item [[John Milnor]], (1965b) \emph{Topology from the Differentiable Viewpoint} (University Press of Virginia) \item Guozhen Wang, Zhouli Xu, \emph{The triviality of the 61-stem in the stable homotopy groups of spheres} (\href{https://arxiv.org /abs/1601.02184}{arXiv:1601.02184}) \item Llohann D. Sperança, \emph{Pulling back the Gromoll-Meyer construction and models of exotic spheres}, Proceedings of the American Mathematical Society 144.7 (2016): 3181-3196 (\href{https://arxiv.org/abs/1010.6039}{arXiv:1010.6039}) \item Llohann D. Sperança, \emph{Explicit Constructions over the Exotic 8-sphere} (\href{https://www.ime.unicamp.br/~rigas/sigma8EncontroTopol.pdf}{pdf}, [[SperancaExoticSpheres.pdf:file]]) \item C. Duran, A. Rigas, Llohann D. Sperança, \emph{Bootstrapping Ad-equivariant maps, diffeomorphisms and involutions}, Matematica Contemporanea, 35:27–39, 2010 (\href{http://www.ime.unicamp.br/~rigas/bootstrapping}{pdf}) \end{itemize} On the open issue of exotic [[4-spheres]]: \begin{itemize}% \item [[Michael Freedman]], [[Robert Gompf]], [[Scott Morrison]], [[Kevin Walker]], \emph{Man and machine thinking about the smooth 4-dimensional Poincar\'e{} conjecture}, Quantum Topology, Volume 1, Issue 2 (2010), pp. 171-208 (\href{http://arxiv.org/abs/0906.5177}{arXiv:0906.5177}) \end{itemize} Review includes \begin{itemize}% \item [[Matthias Kreck]], chapter 19 ``Exotic 7-spheres'' of \emph{Differential Algebraic Topology -- From Stratifolds to Exotic Spheres}, AMS 2010 \item Rachel McEnroe, \emph{Milnor' construction of exotic 7-spheres}, 2015 (\href{http://math.uchicago.edu/~may/REU2015/REUPapers/McEnroe.pdf}{pdf}) \end{itemize} See also \begin{itemize}% \item Wikipedia (\href{http://en.wikipedia.org/wiki/Exotic_sphere#References}{spheres}, \href{http://en.wikipedia.org/wiki/Exotic_R4#References}{$R^4$}) \end{itemize} \hypertarget{ReferencesForApplicationsToPhysics}{}\subsubsection*{{For applications to physics}}\label{ReferencesForApplicationsToPhysics} The relevance of exotic smooth structure to [[physics]] is tantalizing but remains by and large unclear. Some of the following references probably ought to be handled with care. \hypertarget{general_relativity}{}\paragraph*{{General relativity}}\label{general_relativity} The original argument that exotic spheres are to be regarded as [[instantons]] of [[gravity]] is in \begin{itemize}% \item [[Edward Witten]], p. 12 of \emph{Global gravitational anomalies}, Comm. Math. Phys. Volume 100, Number 2 (1985), 197--229. (\href{http://projecteuclid.org/euclid.cmp/1103943444}{EUCLID}) \end{itemize} Further discussion of exotic $4$-manifolds from the [[general relativity]] point of view is in \begin{itemize}% \item [[Carl Brans]], Duane Randall, \emph{Exotic differentiable structures and general relativity} Gen. Rel. Grav., 25 (1993) 205--220 \item [[Carl Brans]] \emph{Exotic smoothness and physics} J. Math. Phys. 35, (1994), 5494--5506. \end{itemize} The following paper contained a first proof to localize exotic smoothness in an exotic $\mathbb{R}^4$: \begin{itemize}% \item [[Carl Brans]] \emph{Localized exotic smoothness} Class. Quant. Grav., 11, (1994), 1785--1792. \end{itemize} A more philosophical discussion can be found in: \begin{itemize}% \item [[Carl Brans]] \emph{Absolute spacetime: the twentieth century ether} Gen. Rel. Grav. 31, (1999), 597--609 \end{itemize} \hypertarget{generation_of_source_terms_fields}{}\paragraph*{{Generation of source terms (fields)}}\label{generation_of_source_terms_fields} Brans conjectured in the papers above, that exotic smoothness should be a source of an additional [[gravity|gravitational field]] (Brans conjecture). This conjecture was confirmed for compact $4$-manifolds (using implicitly a mapping of basic classes): Using the invariant of L. Taylor \href{http://de.arxiv.org/abs/math/9807143}{arXiv}, Sladkowski confirmed the conjecture for the exotic $\mathbb{R}^4$ in: \begin{itemize}% \item Jan Sadkowski \emph{Gravity on exotic R4 with few symmetries} Int.J. Mod. Phys. D, 10, (2001) 311--313 \end{itemize} \hypertarget{quantum_field_theory}{}\paragraph*{{Quantum (field) theory}}\label{quantum_field_theory} The first real connection between exotic smoothness and [[quantum field theory]] is Witten's TQFT: \begin{itemize}% \item [[Edward Witten]], \emph{Topological quantum field theory} Comm. Math. Phys., 117, (1988), 353--386. \end{itemize} and the whole work of Seiberg and Witten leading to the celebrated invariants. The relation to particle physics by using the algebra of smooth functions can be found in \begin{itemize}% \item Jan Sadkowski, \emph{Exotic smoothness, noncommutative geometry and particle physics} Int. J. Theor. Phys., 35, (1996), 2075--2083 \item Jan Sadkowski, \emph{Exotic smoothness and particle physics} Acta Phys. Polon., B 27, (1996), 1649--1652 \item Jan Sadkowski, \emph{Exotic smoothness, fundamental interactions and noncommutative geometry} \href{http://arxiv.org/abs/hep-th/9610093}{arXiv} \end{itemize} The relation between TQFT and differential-topological invariants of smooth manifolds was clarified in: \begin{itemize}% \item [[Hendryk Pfeiffer]] \emph{Quantum general relativity and the classification of smooth manifolds} \href{http://arxiv.org/abs/gr-qc/0404088}{arXiv} \item [[Hendryk Pfeiffer]] \emph{Diffeomorphisms from finite triangulations and absence of `local' degrees of freedom} Phys.Lett. B, 591, (2004), 197-201 \end{itemize} \hypertarget{ReferencesInStringTheory}{}\paragraph*{{String theory}}\label{ReferencesInStringTheory} An argument for interpreting exotic smooth spheres as [[instantons]] of [[gravity]] and to cancel the gravitational anomalies of [[string theory]] is in (\hyperlink{Witten85}{Witten 85}). The influence of exotic smoothness for [[Kaluza-Klein mechanism|Kaluza-Klein models]] was discussed here: \begin{itemize}% \item [[Matthias Kreck]], [[Stefan Stolz]], \emph{A diffeomorphism classification of $7$-dimensional homogeneous Einstein manifolds with $\mathfrak{su}(3) \times \mathfrak{su}(2) \times \mathfrak{u}(1)$-symmetry} Ann. Math. 127, (1988), 373--388. \end{itemize} A discussion of topological effects (also of string theory) in relation to exotic smoothness is in \begin{itemize}% \item Ryan Rohm, \emph{Topological Defects and Differential Structures} Annals Of Physics, 189, (1989), 223--239. \end{itemize} \hypertarget{cosmology}{}\paragraph*{{Cosmology}}\label{cosmology} An overview can be also found in \begin{itemize}% \item Jan Sadkowski, \emph{Exotic smoothness and astrophysics} Act. Phys. Polon. B, 40, (2009), 3157--3163 \end{itemize} \hypertarget{quantum_gravity}{}\paragraph*{{Quantum gravity}}\label{quantum_gravity} A first calculation of the state sum in [[quantum gravity]] by inclusion of exotic smoothness \begin{itemize}% \item Kristin Schleich, Donald Witt, \emph{Exotic Spaces in Quantum Gravity I: Euclidean Quantum Gravity in Seven Dimensions} Class.Quant.Grav., 16, (1999), 2447--2469 \end{itemize} A semi-classical approach to the functional integral is discussed here: \begin{itemize}% \item Christofer Duston, \emph{Exotic smoothness in 4 dimensions and semiclassical Euclidean quantum gravity} \href{http://arxiv.org/abs/0911.4068}{arxiv} \end{itemize} The inclusion of singularities for asymptotically flat spacetimes is discussed here (with an example of a singularity coming from exotic smoothness): \begin{itemize}% \item Kristin Schleich, Donald Witt, \emph{Singularities from the Topology and Differentiable Structure of Asymptotically Flat Spacetimes}, \href{http://arxiv.org/abs/1006.2890}{arxiv} \end{itemize} [[!redirects exotic smooth structure]] [[!redirects exotic smooth structures]] [[!redirects exotic sphere]] [[!redirects exotic spheres]] \end{document}